Temperature-dependent ferroelectric switching of 1–3 type piezocomposites: an experimental and theoretical study

An investigation on the temperature-dependent behavior of 1–3 piezocomposites for different fiber volume fractions and bulk piezoceramics is carried out. Experiments are conducted on 1–3 piezocomposites at room and elevated temperatures under a high cyclic electric field to understand the behavior of these materials. Temperature-dependent effective properties of piezocomposites are measured using resonance technique. A simple analytical model based on equivalent layered approach is proposed for the determination of the effective properties of 1–3 piezocomposites at elevated temperature. A uniaxial constitutive model has been developed based on a thermodynamic approach, to predict the nonlinear behavior of 1–3 piezocomposites under thermo-electrical loading conditions. Volume fractions of three distinct uniaxial variants are used as internal variables to describe the microscopic state of the material. A nonlinear hardening parameter based on a Gaussian function is used to describe the grain boundary effects. The predicted effective properties are used in the proposed uniaxial model and the classical dielectric hysteresis (electric displacement vs. electric field) as well as butterfly curves (strain vs. electric field) are simulated and compared with experimental data. It is observed that the variation in fiber volume fraction and temperature has a strong influence on the response of 1–3 piezocomposites.     

Reliable modeling of piezoceramic materials utilized in sensors and actuators

Piezoceramic materials are widely utilized in actuator and sensor devices. In order to model the behavior of these devices and to reduce their development time, numerical simulation tools are frequently applied. However, the simulation results strongly rely on the material behavior assumed for piezoceramics. Here, we present approaches for reliable modeling of this material behavior which have been developed at the Chair of Sensor Technology (Friedrich-Alexander-University Erlangen-Nuremberg) in recent years. Both the small signal behavior and the large signal behavior of piezoceramic materials are discussed. For the identification of material parameters required within the small signal model, we apply a mathematical Inverse Method. The large signal behavior of piezoceramics is described by means of a phenomenological approach that is based on the so-called Preisach hysteresis operator. As the presented results for different piezoceramics clearly show, the utilized modeling approaches lead to reliable simulation results and can, therefore, be applied to predict the behavior of piezoceramic materials.     

Characterization of the mechanical nonlinear behavior of piezoelectric ceramics

A characterization of the nonlinear behavior with high signal excitation in piezoceramic resonators was carried out. The behavior of power devices working at resonance, in which high strains are involved, is explained. A theoretical model previously described is used to explain the motional impedance variation proportional to the square of the motional current. This impedance increase DeltaZ is independent of the frequency and explains: the nonlinear elasticity that produces the A-F effect, the nonlinear mechanical losses that increase greatly close to the resonance, and the hysteresis phenomenon produced with frequency sweeps. Different methods for measuring the mechanical nonlinear coefficients of piezoceramics with high signal excitation are presented. An experimental method is proposed to measure the mechanical loss tangent and the compliance variations as a function of the mean square strain in the piezoceramic. This consists in measuring the maximum admittance and the series resonance frequency for downward frequencies. At this jumping point, the phase angle remains zero whatever the amplitude of the excitation. Two main coefficients characterizing the material mechanical nonlinearity are deduced. Experimental measurements were carried out to compare the nonlinearity of different ceramic materials in longitudinal and transverse mode.     

A model for the structural hysteresis in poling and thermal depoling of PZT ceramics

A structural hysteresis associated with domain orientation during poling and thermal depoling of lead titanate zirconate (PZT) ceramics has been observed. The poled materials appear to lose their piezoelectric properties at a temperature somewhat below the Curie temperature and yet the domain configurations remain unchanged. The above phenomenon is successfully explained by a model which predicts that upon thermal depolarization, poled ceramics undergo transformation from the poled state into the antiferroelectric state before returning back to their original unpoled state.     

A new simple asymmetric hysteresis operator and its application to inverse control of piezoelectric actuators

Piezoelectric actuators (PEAs) are commonly used as micropositioning devices due to their high resolution, high stiffness, and fast frequency response. Because piezoceramic materials are ferroelectric, they fundamentally exhibit hysteresis behavior in their response to an applied electric field. The positioning precision can be significantly reduced due to nonlinear hysteresis effects when PEAs are used in relatively long range applications. This paper describes a new, precise, and simple asymmetric hysteresis operator dedicated to PEAs. The complex hysteretic transfer characteristic has been considered in a purely phenomenological way, without taking into account the underlying physics. This operator is based on two curves. The first curve corresponds to the main ascending branch and is modeled by the function f1. The second curve corresponds to the main reversal branch and is modeled by the function g2. The functions f(1) and g(2) are two very simple hyperbola functions with only three parameters. Particular ascending and reversal branches are deduced from appropriate translations of f(1) and g(2). The efficiency and precision of the proposed approach is demonstrated, in practice, by a real-time inverse feed-forward controller for piezoelectric actuators. Advantages and drawbacks of the proposed approach compared with classical hysteresis operators are discussed.     

Normally distributed free energy model and creep behavior of ferroelectric polycrystals at room and high temperatures

A constitutive model that can be used to predict creep behavior of ferroelectric polycrystals at room and high temperatures is proposed. The model consists of the Gibbs free energy function with normal distribution and a switching evolution law with critical driving force. Linear moduli in the free energy function and switching parameters in the switching law are assumed to be linearly dependent on temperature. A ferroelectric polycrystal is modeled by an agglomerate of 210 single crystallites. Compressive stress and electric field-induced creep behavior as well as polarization hysteresis and strain butterfly responses of the model are calculated and compared with experimental observations.     

Micro-thermomechanical constitutive model of transformation induced plasticity and its application on armour steel

Based on the crystallographic theory of martensitic transformation and internal variable constitutive theory, a micromechanical constitutive model of martensitic transformation induced plasticity was developed. Plastic strains of product and parent phases as well as the volume fraction of each martensitic variant were considered as internal variables describing the microstructure evolution. The plasticity flow both in austenite and martensitic variants domain is described by J₂ flow theory. The thermodynamic driving force acting on these internal variables was obtained through the determination of the intrinsic dissipation due to plastic flow and the growth of martensitic domains. The evolution laws of the internal variables are derived, furthermore macroscopic response due to the change of internal variables is obtained. Thermomechanical behavior of armour steel under uniaxial loading was tested which showed a good agreement with experimental results.     

A viscoplastic theory with thermodynamic considerations

Summary A thermodynamic foundation using the concept of internal state variables is given for a general theory of viscoplasticity for initially isotropic materials. Three, fundamental, internal, state variables are admitted; they are: a tensorial back stress for kinematic effects, and scalar drag and yield strengths for isotropic effects. All three are considered to evolve phenomenologically according to competitive processes between strain hardening, deformation induced dynamic recovery, and thermally induced static recovery. Within this phenomenological framework, a thermodynamically admissible set of evolution equations is proposed. The theory allows each of the three internal variables to be composed as a sum of independently evolving constituents. The evolution of internal state can also include terms that vary linearly with the external variable rates, whose presence affects the energy dissipation properties of a material.     

An internal state variable approach to constitutive theories for granular materials with snow as an example

In this paper, a microphysical constitutive theory is developed for a class of rate dependent granular materials under finite deformation. The theory is based on non-equilibrium thermodynamics with internal state variables. The state variables may be thought of as representing the current pattern of microstructural arrangemenp and hence characterize the plastic state of the material. A significant feature of this theory is that the state variables are identified at the granular level, as opposed to the crystalline level. This allows one to develop a microdynamical theory in terms of experimentally observable quantities and is a unique feature of granular materials.The theory is used to describe the mechanical properties of snow under high rate multiaxial deformation. Snow is a highly nonlinear, rate dependent material which exhibits significant microstructural alternations under finite strain. These alternations are tracked mathematically by temporal evolution equations governing the internal state variables. The change in the state variables is directly related to the plastic strain of the material.     

On the mathematical modelling of material behavior in continuum mechanics

Summary The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.This paper is an extended version of a lecture held at the First Conference of the GAMM working group on material theory in Stuttgart, Germany, February 28, 1992. The author thanks Prof. Dr. F. Ziegler for the opportunity to participate in this conference.     

Phenomenological modeling of anisotropy induced by evolution of the dislocation structure on the macroscopic and microscopic scale

This work focuses on the modeling of the evolution of anisotropy induced by the development of the dislocation microstructure. A model formulated at the engineering scale in the context of classical metal plasticity and a model formulated in the context of crystal plasticity are presented. Images obtained by transmission-electron microscopy (TEM) show the influence of the strain path on the evolution of anisotropy for the case of two common materials used in sheet metal forming, DC06 and AA6016-T4. Both models are capable of accounting for the transient behavior observed after changes in loading path for fcc and bcc metals. The evolution of the internal variables of the models is correlated with the evolution of the dislocation structure observed by TEM investigations.     

Micromechanical modeling and simulation of rate-dependent effects in ferroelectric polycrystals

In this contribution, rate-dependent switching effects of ferroelectric materials are studied by means of a micromechanically motivated approach. The onset of domain switching is thereby initiated based on the reduction in Gibbs free energy by means of energy-based criterion. The subsequent nucleation and propagation of domain walls during switching process are incorporated via a volume fraction concept combined with a simple linear kinetics theory. The key aspect in modeling of the interaction between the individual grains (intergranular effects) are incorporated in this model by making use of a probabilistic ansatz; to be specific, a phenomenologically motivated Weibull distribution function is adopted. The developed framework is incorporated into a finite element formulation whereby each domain is represented by a single finite element and initial dipole directions are randomly oriented so that the virgin state of the particular bulk ceramics of interest reflects an un-poled material. Based on a staggered iteration technique and straightforward volume averaging concept, the model is simulated to capture the non-linear behavior for different loading, for various loading amplitudes and frequencies. Attributes of the model, both symmetric major loops and biased minor loops are illustrated through examples. Simulation results for the rate-independent case are in good agreement with experimentally measured data reported in the literature and, moreover, are extended to rate-dependent computations which captures some important insights.     

Gradient theory from the thermomechanics point of view

This work demonstrates how gradients of internal state variables can be used in a set of internal variables when the thermomechanics of internal state variables is utilised. This is done by introducing a thermal internal variable called specific dissipative entropy. The gradient term in a material model can be used to avoid localisation. A material model showing Hookean deformation and creep is evaluated. Damage affects the Hookean response. The damage evolution equation contains a Laplacian of damage being introduced to avoid localisation of damage and mesh-dependence of a finite element solution. The material model satisfies the Clausius–Duhem inequality.     

基于不可逆内变量热力学的岩石材料粘弹-粘塑性本构方程

岩石材料的粘弹性和粘塑性变形是与时间相关的能量耗散行为。在Rice不可逆内变量热力学框架下,引入两组内变量分别用来描述在粘弹性和粘塑性变形过程中材料的内部结构调整。通过给定比余能的具体形式和内变量的演化方程,推导出内变量粘弹-粘塑性本构方程。粘弹性本构方程具有普遍性,能涵盖Kelvin-Voigt和Poynting-Thomson在内的经典粘弹性模型的本构方程。并指出热力学力与应力呈线性关系是组合元件模型为线性模型的根本原因。粘塑性本构方程能较好地刻画岩石材料在粘塑性变形过程中的硬化现象。对模拟岩石的模型相似材料进行单轴加卸载蠕变试验,将蠕变过程中的粘弹性和粘塑性变形分离并根据试验数据对本构方程的材料参数进行辨识。试验数据和理论曲线对比结果表明该文提出的本构方程能很好地模拟材料的蠕变行为。该类型的本构方程能为岩石工程的长期稳定性的预测、评价以及加固分析提供基础。     

Effects of scratching on losses in 3-percent Si-Fe single crystals with orientation near

The effects of scratching perpendicular to the [001] direction on losses were studied as a function of the tilt angle of the [001] out of the crystal surface β both with and without tensile stress in 3-percent Si-Fe single crystals. The reduction of the total losses by scratching becomes larger with decreasing β. The total losses of scratched samples further decrease with the application of tensile stress parallel to the [001] direction. Observations of domain structure showed that scratching causes a decrease in 180° main domain wall spacings and also the occurrence of reverse subdomains in the vicinity of the scratch. It was found that the total losses of scratched specimens are lower than those of unscratched samples for equal 180° main domain wall spacing observed in the demagnetized state. This may be because the losses of scratched samples are influenced by the dynamic behavior of reverse subdomains in addition to the function of 180° domain wall displacements, The variation of dc hysteresis loss by scratching is very small. This may be caused by the effect of the new domain configuration at the scratch line, which weakens the domain wall pinning due to free poles or internal stress caused by scratching.     

组合圆筒管的优化设计

用外缠绕玻璃纤维增强复壁钢管来代替油气田内部承高压的高强度钢管,既可节省大量的优质钢材,又能对场内集输管道起到良好的外防腐效果。本文着重研究由玻璃钢/钢组合而成的厚壁圆筒管在管内高压作用下的优化设计问题。首先利用组合管的边界条件,由弹性理论导出了平面应变状态下组合管壁内三个主应力的计算公式;然后建立了以组合管总壁厚、玻璃钢外套管壁厚和玻璃纤维缠绕角为设计变量,单位长度组合管的材料成本为目标函数以及钢管的强度约束和玻璃钢的变形限制为约束条件的优化设计数学模型,采用复形法对本问题求得了最优解。作者还用所编的电算程序计算了各种管内径,各种内压及各种钢管强度等级所对应的设计变量最佳取值,从而讨论了管内径、内压及钢管强度等级变化对优化设计变量的影响,计算实例表明,在承受相同内压作用的情况下,组合管的材料成本较单纯钢管低,组合管的重量减轻尤为明显。主题词:组合圆筒管;内压;平面应变状态;优化设计;复形法。     

Homogenization modeling of domain switching in ferroelectric materials

We presented a multiscale nonlinear finite element simulation to analyze domain switching behaviors in ferroelectric materials. We utilized an incremental form of fundamental constitutive law to consider changes in the material properties caused by domain switching. A multiscale nonlinear problem was formulated by employing the asymptotic homogenization theory based on the perturbation method and implemented using finite element analysis. The developed simulation was applied to barium titanate with a Perovskite-type tetragonal crystal structure. The 90° and 180° domain switching behaviors of a single crystal were computed for verification. The nonlinear behaviors of a bulk polycrystal with virtual microstructure were analyzed as a case study. The variation of the crystal orientation distribution in the polycrystalline microstructure was analyzed to reveal its influence on macroscopic hysteresis and butterfly curves.